Abstract
Usually within the context of computer simulations in quantum chemistry practices (and solid-state physics), there is a distinction between ab initio and semi-empirical methods. Related to this, a controversy within the scientific and philosophical communities came about regarding the superiority of the ab initio methods due to their theoretical rigor. In this article we re-evaluate the condition of the semi-empirical simulations in this area of research. We examine some of the aspects of this debate that have been considered in philosophy and provide additional elements to the analysis.
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Notes
In quantum chemistry, the “basis set” usually refers to the set of (nonorthogonal) one-particle functions used to build molecular orbital. The basis set, in theory, should be complete, but a complete basis set means that an infinite number of functions must be used, which is in fact impossible.
The text refers to the method of molecular orbital developed by Lennard-Jones, Hund, Herzberg and Mulliken; and the strictly homopolar method of Heitler, London, Slater and Pauling.
In general, the Born–Oppenheimer approximation assumes the adiabatic approximation. In this approach, the nuclear motion is confined to the lowest electronic potential surface of the system. There are procedures for which the Born–Oppenheimer approximation is not valid: nonradiative transitions in molecules and solids electron transfer, quenching of excited electronic states, collisional electron excitation, and inelastic electron scattering. “To describe such processes, “nonadiabatic transitions” among different potential-energy surfaces must be accounted for…” (Tully 2000, p. 175).
A paradigmatic example of the tension in the choice of the basis set between a reasonable accuracy and the speed of calculation is the technique developed by Pople and colleagues (Hehre et al. 1969). It consists of replacing each Slater-type orbitals (STO) by a linear combination of small number of Gaussian-type orbitals (GTO’s) to accelerate the calculation, although it implied some loss of accuracy regarding the STO’s. It is worth noting that, currently, the basis set of gaussian orbitals to model the wave functions are the ones that are most frequently used for ab initio simulations that use Hartree–Fock and Post–Hartree–Fock methods.
It’s worth noting that a single basis set is applied to a broad range of molecules, but it does not have the same accuracy for all of them. Also, a chosen basis set does not have the same accuracy for all of the properties for any given molecule.
Designated as p (x, y, z) is one probability per unit of volume. If the charge of an electron is considered to be a unit of charge, then it has electron charge units.
It must be noted that DFT forged the development of numerous hybrid approaches. Special attention deserves to be given to the application of DFT in the field of molecular dynamics that gave way to the formulation of ab initio molecular dynamics (Car and Parrinello 1985), making it possible to investigate large systems. For its importance, this approach deserves a thorough treatment that exceeds the aims of this work.
Even though here we are only referring to external validation, the notion of validation has been widely discussed in the context of computational simulations. In general, these treatments are based on the distinction between internal and external validation. They are said to be externally valid when the computational model represents, to an acceptable degree, the studied realm (Guala 1999). On the other hand, a simulation is internally valid if the solutions of the numerical model approach, to the level of desired accuracy, are relevant solutions to the equations of the original mathematical model (Winsberg 2003). The process of internal validation gives legitimacy to the system, consistency between systems, and does not have any direct implications on the reliability of the model in representing phenomena.
The zero-differential-overlap (ZDO) approximation was the approximation introduced in the initial stages of quantum chemistry to overcome problems in evaluating numerous integrals of three and four centers. This approximation could be addressed at different levels (CNDO, complete neglect of differential overlap; INDO, intermediate neglect of differential overlap; and NDDO, neglect of diatomic differential overlap).
In absence of reliable experimental reference data, accurate theoretical data (e.g., from high-level ab initio calculations) are accepted as substitutes.
MINDO/3: modified intermediate neglect of differential overlap; MNDO: modified neglect of diatomic overlap.
Properties reproduced by MNDO include heats of formation, molecular geometries, dipole moments, ionization energies, electron affinities, polarizabilities, molecular vibration frequencies, thermodynamic properties, kinetic isotope effects and properties of polymers.
There are two kinds of parallel electronic computers. One is shared memory, in which the same machine has multiple processing elements, and the other is distributed memory, where multiple machines are linked to form a cluster of processing elements. Both types enable for the execution of calculations concurrently across multiple processing elements (Needham et al. 2016).
According to Gordon Moore’s projection (1965, 1975), approximately every 2 years the number of transistors in a microprocessor will double, until reaching the limit of what is physically possible. Likewise, David House predicted that it would take 18 months for the same increase, taking into consideration their speed as well. “… the landscape of computer hardware is constantly changing, but not all changes are equally disruptive to scientific software programmers. After approximately three decades of continuous performance improvement from steadily increasing clock frequencies and concomitant decrease in semiconductor feature size, microprocessor design underwent a fundamental shift toward parallelism at the beginning of the last decade. Having reached the limits of frequency-scaling due to power constraints, node-level parallelism is now the basis for keeping pace with Moore’s law.” (DePrince III et al. 2016, pp. 279–280).
Furthermore, Kohn highlight that, even if there were no computational limits, other limiting factors emerge when the number of electrons increases and correcting for these physical factors, such as relativistic or radioactive, would be limiting (Kohn 1999).
With semi-empirical simulations, one can deal with systems containing thousands of atoms. An improvement of the conventional semi-empirical method allows for executing energy calculations with high accuracy on molecules that have thousands of atoms: polyglycine chains of 20,000 atoms; a group of water molecules with more than 1800 atoms, and nucleic acids of more than 6300 atoms (Daniels et al. 1997; Wu et al. 2016).
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Acknowledgements
We would like to thank Professor Victor Rodríguez who provided insight and expertise that greatly assisted the research. In other respect, this research was supported in part by the Fund for Scientific and Technological Research (FONCYT) PICT-2016-1524.
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Polzella, M.S., Lodeyro, P. Re-evaluating semi-empirical computer simulations in quantum chemistry. Found Chem 21, 83–95 (2019). https://doi.org/10.1007/s10698-018-09329-w
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DOI: https://doi.org/10.1007/s10698-018-09329-w